Question

For what values of x, if any, does `f(x) = cot((7pi)/12-x)` have vertical asymptotes?

Answer 1

`x= (7pi)/12 - 2pin`
`x = -(5pi)/12 - 2pin`

For `n in ZZ`

Explanation:

To approach this we must cosnider how `cotx = cosx/sinx`

Hence for `cotx` to have verticle asymptotes, the denominator must be 0, so hence `sinx = 0`

So hence in this circumstance ;
`sin((7pi)/12 - x) = 0` for the verticle asymptotes

So hence now we consider the general solution of;
`sinx = sina`
Then `x = 2pin + a`
and `x = 2pin + pi -a`

So hence `sin( (7pi) /12 - x) = sin0`

Hence; `(7pi)/12 - x = 2pin`
and `(7pi)/12 - x =2pin + pi`

Hence rearanging to yeild;

`x= (7pi)/12 - 2pin`
`x = -(5pi)/12 - 2pin`

For `n in ZZ`

We can consider this by graphing;
graph{cot( 7pi/12 -x) [-4.354, 5.644, -2.28, 2.72]}